Trowbridge-Reitz Sample, near normal, slope_y

Percentage Accurate: 98.3% → 98.4%

Time: 10.8s

Alternatives: 11

Speedup: 1.0×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

Initial Program: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((6.28318530718f * u2));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin((6.28318530718e0 * u2))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(Float32(6.28318530718) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((single(6.28318530718) * u2));
end

Alternative 1: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (sqrt (* (pow u2 2.0) 39.47841760436263)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(sqrtf((powf(u2, 2.0f) * 39.47841760436263f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin(sqrt(((u2 ** 2.0e0) * 39.47841760436263e0)))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(sqrt(Float32((u2 ^ Float32(2.0)) * Float32(39.47841760436263)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin(sqrt(((u2 ^ single(2.0)) * single(39.47841760436263))));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 97.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{6.28318530718 \cdot u2} \cdot \sqrt{6.28318530718 \cdot u2}\right)} \]

   2. sqrt-unprod 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{\left(6.28318530718 \cdot u2\right) \cdot \left(6.28318530718 \cdot u2\right)}\right)} \]

   3. *-commutative 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\color{blue}{\left(u2 \cdot 6.28318530718\right)} \cdot \left(6.28318530718 \cdot u2\right)}\right) \]

   4. *-commutative 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\left(u2 \cdot 6.28318530718\right) \cdot \color{blue}{\left(u2 \cdot 6.28318530718\right)}}\right) \]

   5. swap-sqr 98.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\color{blue}{\left(u2 \cdot u2\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)}}\right) \]

   6. pow2 98.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{\color{blue}{{u2}^{2}} \cdot \left(6.28318530718 \cdot 6.28318530718\right)}\right) \]

   7. metadata-eval 98.5%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\sqrt{{u2}^{2} \cdot \color{blue}{39.47841760436263}}\right) \]

4. Applied egg-rr 98.5%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)} \]
5. Add Preprocessing

Alternative 2: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(u2 \cdot 6.28318530718\right)\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (expm1 (log1p (* u2 6.28318530718))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf(expm1f(log1pf((u2 * 6.28318530718f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(expm1(log1p(Float32(u2 * Float32(6.28318530718))))))
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. expm1-log1p-u 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot u2\right)\right)\right)} \]

   2. expm1-undefine 60.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(e^{\mathsf{log1p}\left(6.28318530718 \cdot u2\right)} - 1\right)} \]

4. Applied egg-rr 60.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(e^{\mathsf{log1p}\left(6.28318530718 \cdot u2\right)} - 1\right)} \]
5. Step-by-step derivation

   1. expm1-define 98.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot u2\right)\right)\right)} \]

6. Simplified 98.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(6.28318530718 \cdot u2\right)\right)\right)} \]

7. Final simplification 98.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(u2 \cdot 6.28318530718\right)\right)\right) \]
8. Add Preprocessing

Alternative 3: 93.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004550000187009573:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{u1} \cdot \frac{u2}{\sqrt{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.004550000187009573)
   (* 6.28318530718 (* (sqrt u1) (/ u2 (sqrt (- 1.0 u1)))))
   (* (sin (* u2 6.28318530718)) (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.004550000187009573f) {
		tmp = 6.28318530718f * (sqrtf(u1) * (u2 / sqrtf((1.0f - u1))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf((u1 * (u1 + 1.0f)));
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.004550000187009573e0) then
        tmp = 6.28318530718e0 * (sqrt(u1) * (u2 / sqrt((1.0e0 - u1))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt((u1 * (u1 + 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.004550000187009573))
		tmp = Float32(Float32(6.28318530718) * Float32(sqrt(u1) * Float32(u2 / sqrt(Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(Float32(u1 * Float32(u1 + Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.004550000187009573))
		tmp = single(6.28318530718) * (sqrt(u1) * (u2 / sqrt((single(1.0) - u1))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt((u1 * (u1 + single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00455000019

   1. Initial program 98.6%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0 97.7%

      \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 97.7%

         \[\leadsto 6.28318530718 \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. sqrt-div 97.4%

         \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \color{blue}{\frac{\sqrt{u1}}{\sqrt{1 - u1}}}\right) \]

      3. associate-*r/ 97.4%

         \[\leadsto 6.28318530718 \cdot \color{blue}{\frac{u2 \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

   5. Applied egg-rr 97.4%

      \[\leadsto 6.28318530718 \cdot \color{blue}{\frac{u2 \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]
   6. Step-by-step derivation

      1. *-commutative 97.4%

         \[\leadsto 6.28318530718 \cdot \frac{\color{blue}{\sqrt{u1} \cdot u2}}{\sqrt{1 - u1}} \]

      2. associate-/l* 97.7%

         \[\leadsto 6.28318530718 \cdot \color{blue}{\left(\sqrt{u1} \cdot \frac{u2}{\sqrt{1 - u1}}\right)} \]

   7. Simplified 97.7%

      \[\leadsto 6.28318530718 \cdot \color{blue}{\left(\sqrt{u1} \cdot \frac{u2}{\sqrt{1 - u1}}\right)} \]

   if 0.00455000019 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0 86.8%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   4. Step-by-step derivation

      1. +-commutative 86.8%

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

   5. Simplified 86.8%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.004550000187009573:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{u1} \cdot \frac{u2}{\sqrt{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.005200000014156103:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{u1} \cdot \frac{u2}{\sqrt{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.005200000014156103)
   (* 6.28318530718 (* (sqrt u1) (/ u2 (sqrt (- 1.0 u1)))))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.005200000014156103f) {
		tmp = 6.28318530718f * (sqrtf(u1) * (u2 / sqrtf((1.0f - u1))));
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.005200000014156103e0) then
        tmp = 6.28318530718e0 * (sqrt(u1) * (u2 / sqrt((1.0e0 - u1))))
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.005200000014156103))
		tmp = Float32(Float32(6.28318530718) * Float32(sqrt(u1) * Float32(u2 / sqrt(Float32(Float32(1.0) - u1)))));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.005200000014156103))
		tmp = single(6.28318530718) * (sqrt(u1) * (u2 / sqrt((single(1.0) - u1))));
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00520000001

   1. Initial program 98.6%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0 97.5%

      \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
   4. Step-by-step derivation

      1. *-commutative 97.5%

         \[\leadsto 6.28318530718 \cdot \color{blue}{\left(u2 \cdot \sqrt{\frac{u1}{1 - u1}}\right)} \]

      2. sqrt-div 97.2%

         \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \color{blue}{\frac{\sqrt{u1}}{\sqrt{1 - u1}}}\right) \]

      3. associate-*r/ 97.3%

         \[\leadsto 6.28318530718 \cdot \color{blue}{\frac{u2 \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]

   5. Applied egg-rr 97.3%

      \[\leadsto 6.28318530718 \cdot \color{blue}{\frac{u2 \cdot \sqrt{u1}}{\sqrt{1 - u1}}} \]
   6. Step-by-step derivation

      1. *-commutative 97.3%

         \[\leadsto 6.28318530718 \cdot \frac{\color{blue}{\sqrt{u1} \cdot u2}}{\sqrt{1 - u1}} \]

      2. associate-/l* 97.5%

         \[\leadsto 6.28318530718 \cdot \color{blue}{\left(\sqrt{u1} \cdot \frac{u2}{\sqrt{1 - u1}}\right)} \]

   7. Simplified 97.5%

      \[\leadsto 6.28318530718 \cdot \color{blue}{\left(\sqrt{u1} \cdot \frac{u2}{\sqrt{1 - u1}}\right)} \]

   if 0.00520000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0 76.2%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.005200000014156103:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{u1} \cdot \frac{u2}{\sqrt{1 - u1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.005200000014156103:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 6.28318530718) 0.005200000014156103)
   (* 6.28318530718 (* (sqrt (/ u1 (- 1.0 u1))) u2))
   (* (sin (* u2 6.28318530718)) (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * 6.28318530718f) <= 0.005200000014156103f) {
		tmp = 6.28318530718f * (sqrtf((u1 / (1.0f - u1))) * u2);
	} else {
		tmp = sinf((u2 * 6.28318530718f)) * sqrtf(u1);
	}
	return tmp;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    real(4) :: tmp
    if ((u2 * 6.28318530718e0) <= 0.005200000014156103e0) then
        tmp = 6.28318530718e0 * (sqrt((u1 / (1.0e0 - u1))) * u2)
    else
        tmp = sin((u2 * 6.28318530718e0)) * sqrt(u1)
    end if
    code = tmp
end function

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(6.28318530718)) <= Float32(0.005200000014156103))
		tmp = Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2));
	else
		tmp = Float32(sin(Float32(u2 * Float32(6.28318530718))) * sqrt(u1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
	if ((u2 * single(6.28318530718)) <= single(0.005200000014156103))
		tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * u2);
	else
		tmp = sin((u2 * single(6.28318530718))) * sqrt(u1);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 #s(literal 314159265359/50000000000 binary32) u2) < 0.00520000001

   1. Initial program 98.6%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u2 around 0 97.5%

      \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]

   if 0.00520000001 < (*.f32 #s(literal 314159265359/50000000000 binary32) u2)

   1. Initial program 97.7%

      \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in u1 around 0 76.2%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sin \left(6.28318530718 \cdot u2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot 6.28318530718 \leq 0.005200000014156103:\\ \;\;\;\;6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (/ u1 (- 1.0 u1))) (sin (* u2 6.28318530718))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 / (1.0f - u1))) * sinf((u2 * 6.28318530718f));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 / (1.0e0 - u1))) * sin((u2 * 6.28318530718e0))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * sin(Float32(u2 * Float32(6.28318530718))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 / (single(1.0) - u1))) * sin((u2 * single(6.28318530718)));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Final simplification 98.3%

   \[\leadsto \sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(u2 \cdot 6.28318530718\right) \]
4. Add Preprocessing

Alternative 7: 81.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* (sqrt (/ u1 (- 1.0 u1))) u2)))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (sqrtf((u1 / (1.0f - u1))) * u2);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (sqrt((u1 / (1.0e0 - u1))) * u2)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(sqrt(Float32(u1 / Float32(Float32(1.0) - u1))) * u2))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (sqrt((u1 / (single(1.0) - u1))) * u2);
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 81.8%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]
4. Add Preprocessing

Alternative 8: 73.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt (* u1 (+ u1 1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * sqrtf((u1 * (u1 + 1.0f))));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * sqrt((u1 * (u1 + 1.0e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(Float32(u1 * Float32(u1 + Float32(1.0))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt((u1 * (u1 + single(1.0)))));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 81.8%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]

4. Taylor expanded in u1 around 0 73.5%

   \[\leadsto 6.28318530718 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(1 + u1\right)}} \cdot u2\right) \]
5. Step-by-step derivation

   1. +-commutative 86.6%

      \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 + 1\right)}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]

6. Simplified 73.5%

   \[\leadsto 6.28318530718 \cdot \left(\sqrt{\color{blue}{u1 \cdot \left(u1 + 1\right)}} \cdot u2\right) \]

7. Final simplification 73.5%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1 \cdot \left(u1 + 1\right)}\right) \]
8. Add Preprocessing

Alternative 9: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* u2 6.28318530718) (sqrt u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return (u2 * 6.28318530718f) * sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (u2 * 6.28318530718e0) * sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(u2 * Float32(6.28318530718)) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (u2 * single(6.28318530718)) * sqrt(u1);
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 81.8%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]

4. Taylor expanded in u1 around 0 63.9%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 63.8%

      \[\leadsto \color{blue}{\sqrt{6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)} \cdot \sqrt{6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)}} \]

   2. sqrt-unprod 63.9%

      \[\leadsto \color{blue}{\sqrt{\left(6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)\right) \cdot \left(6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)\right)}} \]

   3. *-commutative 63.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot 6.28318530718\right)} \cdot \left(6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)\right)} \]

   4. *-commutative 63.9%

      \[\leadsto \sqrt{\left(\left(\sqrt{u1} \cdot u2\right) \cdot 6.28318530718\right) \cdot \color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot 6.28318530718\right)}} \]

   5. swap-sqr 63.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{u1} \cdot u2\right) \cdot \left(\sqrt{u1} \cdot u2\right)\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)}} \]

   6. swap-sqr 63.9%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{u1} \cdot \sqrt{u1}\right) \cdot \left(u2 \cdot u2\right)\right)} \cdot \left(6.28318530718 \cdot 6.28318530718\right)} \]

   7. add-sqr-sqrt 64.0%

      \[\leadsto \sqrt{\left(\color{blue}{u1} \cdot \left(u2 \cdot u2\right)\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)} \]

   8. unpow2 64.0%

      \[\leadsto \sqrt{\left(u1 \cdot \color{blue}{{u2}^{2}}\right) \cdot \left(6.28318530718 \cdot 6.28318530718\right)} \]

   9. metadata-eval 63.8%

      \[\leadsto \sqrt{\left(u1 \cdot {u2}^{2}\right) \cdot \color{blue}{39.47841760436263}} \]

6. Applied egg-rr 63.8%

   \[\leadsto \color{blue}{\sqrt{\left(u1 \cdot {u2}^{2}\right) \cdot 39.47841760436263}} \]
7. Step-by-step derivation

   1. associate-*l* 63.9%

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left({u2}^{2} \cdot 39.47841760436263\right)}} \]

   2. sqrt-prod 63.8%

      \[\leadsto \color{blue}{\sqrt{u1} \cdot \sqrt{{u2}^{2} \cdot 39.47841760436263}} \]

   3. *-commutative 63.8%

      \[\leadsto \sqrt{u1} \cdot \sqrt{\color{blue}{39.47841760436263 \cdot {u2}^{2}}} \]

   4. metadata-eval 63.9%

      \[\leadsto \sqrt{u1} \cdot \sqrt{\color{blue}{\left(6.28318530718 \cdot 6.28318530718\right)} \cdot {u2}^{2}} \]

   5. metadata-eval 63.8%

      \[\leadsto \sqrt{u1} \cdot \sqrt{\left(\color{blue}{\sqrt{39.47841760436263}} \cdot 6.28318530718\right) \cdot {u2}^{2}} \]

   6. unpow1/2 63.8%

      \[\leadsto \sqrt{u1} \cdot \sqrt{\left(\color{blue}{{39.47841760436263}^{0.5}} \cdot 6.28318530718\right) \cdot {u2}^{2}} \]

   7. metadata-eval 63.6%

      \[\leadsto \sqrt{u1} \cdot \sqrt{\left({39.47841760436263}^{0.5} \cdot \color{blue}{\sqrt{39.47841760436263}}\right) \cdot {u2}^{2}} \]

   8. unpow1/2 63.6%

      \[\leadsto \sqrt{u1} \cdot \sqrt{\left({39.47841760436263}^{0.5} \cdot \color{blue}{{39.47841760436263}^{0.5}}\right) \cdot {u2}^{2}} \]

   9. unpow2 63.6%

      \[\leadsto \sqrt{u1} \cdot \sqrt{\left({39.47841760436263}^{0.5} \cdot {39.47841760436263}^{0.5}\right) \cdot \color{blue}{\left(u2 \cdot u2\right)}} \]

   10. swap-sqr 63.7%

       \[\leadsto \sqrt{u1} \cdot \sqrt{\color{blue}{\left({39.47841760436263}^{0.5} \cdot u2\right) \cdot \left({39.47841760436263}^{0.5} \cdot u2\right)}} \]

   11. sqrt-unprod 63.6%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(\sqrt{{39.47841760436263}^{0.5} \cdot u2} \cdot \sqrt{{39.47841760436263}^{0.5} \cdot u2}\right)} \]

   12. add-sqr-sqrt 63.7%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left({39.47841760436263}^{0.5} \cdot u2\right)} \]

   13. *-commutative 63.7%

       \[\leadsto \sqrt{u1} \cdot \color{blue}{\left(u2 \cdot {39.47841760436263}^{0.5}\right)} \]

   14. unpow1/2 63.7%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{\sqrt{39.47841760436263}}\right) \]

   15. metadata-eval 63.9%

       \[\leadsto \sqrt{u1} \cdot \left(u2 \cdot \color{blue}{6.28318530718}\right) \]

8. Applied egg-rr 63.9%

   \[\leadsto \color{blue}{\sqrt{u1} \cdot \left(u2 \cdot 6.28318530718\right)} \]

9. Final simplification 63.9%

   \[\leadsto \left(u2 \cdot 6.28318530718\right) \cdot \sqrt{u1} \]
10. Add Preprocessing

Alternative 10: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* 6.28318530718 (* u2 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return 6.28318530718f * (u2 * sqrtf(u1));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 6.28318530718e0 * (u2 * sqrt(u1))
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(6.28318530718) * Float32(u2 * sqrt(u1)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(6.28318530718) * (u2 * sqrt(u1));
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0 81.8%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot u2\right)} \]

4. Taylor expanded in u1 around 0 63.9%

   \[\leadsto \color{blue}{6.28318530718 \cdot \left(\sqrt{u1} \cdot u2\right)} \]

5. Final simplification 63.9%

   \[\leadsto 6.28318530718 \cdot \left(u2 \cdot \sqrt{u1}\right) \]
6. Add Preprocessing

Alternative 11: 7.1% accurate, 209.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 0.0)

C

float code(float cosTheta_i, float u1, float u2) {
	return 0.0f;
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = 0.0e0
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(0.0)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = single(0.0);
end

Derivation

1. Initial program 98.3%

   \[\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 94.4%

      \[\leadsto \color{blue}{\sqrt{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)} \cdot \sqrt{\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)}} \]

   2. sqrt-unprod 95.1%

      \[\leadsto \color{blue}{\sqrt{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right) \cdot \left(\sqrt{\frac{u1}{1 - u1}} \cdot \sin \left(6.28318530718 \cdot u2\right)\right)}} \]

   3. swap-sqr 95.0%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt{\frac{u1}{1 - u1}} \cdot \sqrt{\frac{u1}{1 - u1}}\right) \cdot \left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)}} \]

   4. add-sqr-sqrt 95.2%

      \[\leadsto \sqrt{\color{blue}{\frac{u1}{1 - u1}} \cdot \left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)} \]

   5. pow2 95.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{{\sin \left(6.28318530718 \cdot u2\right)}^{2}}} \]

4. Applied egg-rr 95.2%

   \[\leadsto \color{blue}{\sqrt{\frac{u1}{1 - u1} \cdot {\sin \left(6.28318530718 \cdot u2\right)}^{2}}} \]
5. Step-by-step derivation

   1. unpow2 95.2%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(\sin \left(6.28318530718 \cdot u2\right) \cdot \sin \left(6.28318530718 \cdot u2\right)\right)}} \]

   2. sqr-sin-a 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(6.28318530718 \cdot u2\right)\right)\right)}} \]

   3. metadata-eval 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\color{blue}{\sqrt{39.47841760436263}} \cdot u2\right)\right)\right)} \]

   4. pow1 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\sqrt{39.47841760436263} \cdot \color{blue}{{u2}^{1}}\right)\right)\right)} \]

   5. metadata-eval 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\sqrt{39.47841760436263} \cdot {u2}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)\right)\right)} \]

   6. sqrt-pow1 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\sqrt{39.47841760436263} \cdot \color{blue}{\sqrt{{u2}^{2}}}\right)\right)\right)} \]

   7. sqrt-prod 37.5%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \color{blue}{\sqrt{39.47841760436263 \cdot {u2}^{2}}}\right)\right)} \]

   8. *-commutative 37.5%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \sqrt{\color{blue}{{u2}^{2} \cdot 39.47841760436263}}\right)\right)} \]

   9. sqr-sin-a 95.3%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(\sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right) \cdot \sin \left(\sqrt{{u2}^{2} \cdot 39.47841760436263}\right)\right)}} \]

   10. sin-mult 37.5%

       \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\frac{\cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263} - \sqrt{{u2}^{2} \cdot 39.47841760436263}\right) - \cos \left(\sqrt{{u2}^{2} \cdot 39.47841760436263} + \sqrt{{u2}^{2} \cdot 39.47841760436263}\right)}{2}}} \]

6. Applied egg-rr 37.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\frac{\cos \left(u2 \cdot 6.28318530718 - u2 \cdot 6.28318530718\right) - \cos \left(u2 \cdot 6.28318530718 + u2 \cdot 6.28318530718\right)}{2}}} \]
7. Step-by-step derivation

   1. div-sub 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(\frac{\cos \left(u2 \cdot 6.28318530718 - u2 \cdot 6.28318530718\right)}{2} - \frac{\cos \left(u2 \cdot 6.28318530718 + u2 \cdot 6.28318530718\right)}{2}\right)}} \]

   2. +-inverses 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(u2 \cdot 6.28318530718 + u2 \cdot 6.28318530718\right)}{2}\right)} \]

   3. cos-0 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(\frac{\color{blue}{1}}{2} - \frac{\cos \left(u2 \cdot 6.28318530718 + u2 \cdot 6.28318530718\right)}{2}\right)} \]

   4. metadata-eval 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(\color{blue}{0.5} - \frac{\cos \left(u2 \cdot 6.28318530718 + u2 \cdot 6.28318530718\right)}{2}\right)} \]

   5. distribute-lft-out 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - \frac{\cos \color{blue}{\left(u2 \cdot \left(6.28318530718 + 6.28318530718\right)\right)}}{2}\right)} \]

   6. metadata-eval 37.4%

      \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - \frac{\cos \left(u2 \cdot \color{blue}{12.56637061436}\right)}{2}\right)} \]

8. Simplified 37.4%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \color{blue}{\left(0.5 - \frac{\cos \left(u2 \cdot 12.56637061436\right)}{2}\right)}} \]

9. Taylor expanded in u2 around 0 7.1%

   \[\leadsto \sqrt{\frac{u1}{1 - u1} \cdot \left(0.5 - \color{blue}{0.5}\right)} \]

10. Taylor expanded in u1 around 0 7.1%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024145 
(FPCore (cosTheta_i u1 u2)
  :name "Trowbridge-Reitz Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (/ u1 (- 1.0 u1))) (sin (* 6.28318530718 u2))))